The Γ-limit and the related gradient flow for singular perturbation functionals of Perona-Malik type
Giovanni Bellettini, Giorgio Fusco · Transactions of the American Mathematical Society · 2008
We consider a class of nonconvex functionals of the gradient in one dimension, which we regularize with a second order derivative term. After a proper rescaling, suggested by the associated dynamical problems, we show that the sequence { F ν } \{F_ u \} of regularized functionals Γ \Gamma -converges, as ν → 0 + u \to 0^+ , to a particular class of free-discontinuity functionals F \mathcal {F} , concentrated on S B V SBV functions with finite energy and having only the jump part in the derivative. We study the singular dynamic associated with F \mathcal {F} , using the minimizing movements method. We show that the minimizing movement starting from an initial datum with a finite number of discontinuities has jump positions fixed in space and whose number is nonincreasing with time. Moreover, there are a finite number of singular times at which there is a dropping of the number of discontinuities. In the interval between two subsequent singular times, the vector of the survived jumps is determined by the system of ODEs which expresses the L 2 L^2 -gradient of the Γ \Gamma -limit. Furthermore the minimizing movement turns out to be continuous with respect to the initial datum. Some properties of a minimizing movement starting from a function with an infinite number of discontinuities are also derived.